BULLETIN of the Bull . Malays . Math . Sci . Soc .(2)30(1)(2007), 37endash − f our2 Malaysian Mathematical
Sciences Society http : / / math . usm . my / bullet i n
On 2 - Quasi - Umbilical Pseudosymmetric Hypersurfaces
in
the Euclidean Space
Cihan ¨Ozg ¨ur
Balikesir University , Faculty of Art and Sciences , Department of Mathematics , 10 145 Balikesir , Turkey
cozgur @balikesir . edu . tr
Abstract . In this paper , we investigate 2 quasi umbilical pseudosymmetric hy -persurfaces in the Euclidean space En+1.We find the curvature characterization
of pseudosymmetric hypersurfaces in the Euclidean space En + 1.
2000 Mathematics Subject Classification : 53 C 40 , 53 C 42 , 53 C 25 , 53 B 50 Key words and phrases : Hypersurface , pseudosymmetry type manifolds . 1 . Introduction
Let (M, g) be an n− dimensional , n ≥ 3, connected Riemannian manifold of class
C ∞
. W e denote by ∇, R, C, S and κ the Levi Civita connection , the Riemann -Christoffel curvature tensor , the Weyl conformal curvature tensor , the Ricci tensor and the scalar curvature of (M, g) respectively . The Ricci operator S is defined by g(SX, Y ) = S(X, Y ), where X, Y ∈ χ(M ), χ(M ) being Lie algebra of vector fields on M. We next define endomorphisms X ∧ Y, R(X, Y ) and C(X, Y ) of χ(M ) by
(X ∧ Y )Z = g(Y, Z)X − g(X, Z)Y, (1.1) R(X, Y )Z = ∇X∇YZ − ∇Y∇XZ − ∇[X,Y ]Z, (1.2) C(X, Y )Z = R(X, Y )Z − 1 n − 2(X ∧ SY + SX ∧ Y − κ n − 1X ∧ Y )Z, (1.3) respectively, whereX, Y, Z ∈ χ(M ).
The Riemannian Christoffel curvature tensor R and the Weyl conformal curvature tensor C of (M, g) are defined by
R(X, Y, Z, W ) = g(R(X, Y )Z, W ), (1.4) C(X, Y, Z, W ) = g(C(X, Y )Z, W ), (1.5)
respectively, whereW ∈ χ(M ).
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For a (0, k)− tensor field T, k ≥ 1, on (M, g) we define the tensors R.T and Q(g, T ) by (R(X, Y ) · T )(X1, ..., Xk) = −T (R(X, Y )X1, X2, ..., Xk) − · · · −T (X1, ..., Xk−1, R(X, Y )Xk), (1.6) Q(g, T )(X1, ..., Xk; X, Y ) = −T ((X ∧ Y )X1, X2, ..., Xk) − · · · −T (X1, ..., Xk−1, (X ∧ Y )Xk), (1.7) respectively .
If the tensors R · R and Q(g, R) are linearly dependent then M is called pseu -dosymmetric . This is equivalent to
R · R = LRQ(g, R) (1.8)
holding on the set UR = {x | Q(g, R) 6= 0 at x}, where LR is some function on UR. If
R · R = 0 then M is called s emisymmetric ( see Deszcz [ 3 ] ) .
If the tensors R · S and Q(g, S) are linearly dependent then M is called Ricci -pseudosymmetric . This is equivalent to
R · S = LSQ(g, S) (1.9)
holding on the set US = {x | S 6= κng at x}, where LS is some function on US. Every
pseudosymmetric manifold is Ricci pseudosymmetric but the converse statement is not true . If R · S = 0 then M is called Ricci - s emisymmetric ( see Deszcz [ 3 ] ) .
If the tensors R · C and Q(g, C) are linearly dependent then M is called Weyl -pseudosymmetric . This is equivalent to
R · C = LCQ(g, C) (1.10)
holding on the set UC = {x | C 6= 0 at x}. Every pseudosymmetric manifold is Weyl
pseudosymmetric but the converse st atement is not true . If R · C = 0 then M is called Weyl - s emisymmetric ( see Deszcz [ 3 ] ) .
The manifold M is a manifold with pseudosymmetric Weyl t ensor if and only if
C · C = LCQ(g, C) (1.11)
holds on the set UC, where LC is some function on UC( see Deszcz , Verstraelen , and
Yaprak , [ 4 ] ) . The tensor C · C is defined in the same way as the tensor R · R. 2 . 2 - quasi umbilical hypersurfaces
Let Mn be an n ≥ 3 dimensional connected hypersurface immersed isometrically in the
Euclidean space En+1. We denote by e∇ and ∇ the Levi - Civita connections
corresponding to En+1 and Mn, respectively . Let ξ be a lo cal unit normal vector field
on Mn
in En+1. We can present the Gauss formula and the Weingarten formula of Mn
in En+1 of the form :
e
∇XY = ∇XY + h(X, Y ) and e∇Xξ = −Aξ(X)+
DXξ, respectively , where X, Y are vector fields tangent to Mn and D is the normal
For the plane section ei∧ ej of the tangent bundle T Mn spanned by the vectors
n
ei and ej(i 6= j) the scalar curvature of Mn is defined by κ = P K(ei∧ ej) where
i, j = 1
K denotes the sectional curvature of Mn. We denote by shortly K
On 2 - Quasi - Umbilical Pseudosymmetric Hypersurfaces i − nh − te Euclide a − nSpace 39
Hypersurface Mn with three distinct principal curvatures , their multiplicities are 1 , 1 and n − 2, is said to be 2 - quasi umbilical . So the shape operator of a 2 - quasi - umbilical hypersurface is of the form
Aξ= a 0 0 0 · · · 0 0 b 0 0 · · · 0 0 0 c 0 · · · 0 0 0 0 c · · · 0 . . . . . . . . . . . . 0 0 0 0 · · · c . (2.1)
2 - quasi - umbilical hypersurfaces are the extended class of quasi - umbilical hyper - surfaces . It is well - known that a hypersurface Mn which has a principal curvature
with multiplicity ≥ n − 1 is said to be quasi - umbilical . The well - known result of E . Cartan gives us “ A hypersurface Mn, n ≥ 4, isometrically in the Euclidean space
En+1, is conformally flat if and only if it is quasi umbilical
00
. In ¨Ozgu¨r[8], the present author studied conformally flat submanifolds with flat normal connection .
By ( 2 . 1 ) for a 2 - quasi - umbilical hypersurface , one can get easily the following
corollaries :
Corollary 2 . 1 . Let Mn be a 2 - quasi - umbilical hypersurface of
En+1, n ≥ 4, then K12= ab, KK1j 2j ==acbc , , ((jj> >22) ) (2.2) Kij = c2, (i, j > 2).
where i, j > 2. Furthermore , R(ei, ej)ek= 0 if i, j and k are mutually different .
Theorem 2 . 1 . [ 5 ] . Any 2 - quasi - umbilical hypersurface Mn, dimMn≥ 4, immersed
is ometrically in a s emi - Riemannian conformally flat manifold N is a manifold with pseudosymmetric Weyl t ensor .
On the other hand , it is known that in a hypersurface Mn of a Riemannian space of
constant curvature Nn+1(c), n ≥ 4, if Mn is a Ricci - pseudosymmetric manifold with
pseudosymmetric Weyl tensor then it is a pseudosymmetric manifold ( see Deszcz , Verstraelen , and Yaprak [ 4 ] ) . Moreover from Arslan , Deszcz , and Yaprak [ 1 ] , we know that , in a hypersurface Mn of a Riemannian space of constant
curvature
Nn+1(c), n ≥ 4, the Weyl pseudosymmetry and the pseudosymmetry conditions are
equivalent . So using the previous facts and Theorem 2 . 1 one can obtain the following corollary .
Corollary 2 . 2 . In the class of 2 - quasi - umbilical hypersurfaces of th e Euclidean space
En+1, n ≥ 4, the conditions of th e pseudosymmetry , Ricci pseudosymmetry and Weyl pseudosymmetry are equivalent .
In ¨Ozgu¨r and Arslan [ 9 ] , the present author and K . Arslan studied pseudosym
-metry type hypersurfaces in the Euclidean space satisfying Chen ’ s equality . It is known that , a hypersurface satisfying Chen ’ s equality is a special 2 - quasi - umbilical hypersurface .
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In this study , our aim is to generalize the study ¨Ozgu¨r and Arslan [ 9 ] and to
find the characterization of 2 quasi umbilical hypersurfaces satisfying pseudosymme -try curvature condition . Since pseudosymmetry , Ricci - pseudosymmetry and Weyl - pseudosymmetry curvature conditions for a 2 - quasi - umbilical hypersurface in the Euclidean space En+1, are equivalent , it is sufficient to investigate only pseudosym
-metry curvature condition .
Firstly we have :
Lemma 2 . 1 . Let Mn be a 2 - quasi - umbilical hypersurface of
En+1, n ≥ 4. Then
(R(e1, e3) · R)(e2, e3)e1= abc[c − a]e2, (2.3)
(R(e2, e3) · R)(e1, e3)e2= abc[c − b]e1. (2.4)
Proof . Using ( 1 . 6 ) we have
(R(e1, e3) · R)(e2, e3)e1
= R(e1, e3)(R(e2, e3)e1) − R(R(e1, e3)e2, e3)e1 (2.5)
−R(e2, R(e1, e3)e3)e1− R(e2, e3)(R(e1, e3)e1)
and
(R(e2, e3) · R)(e1, e3)e2
= R(e2, e3)(R(e1, e3)e2) − R(R(e2, e3)e1, e3)e2 (2.6)
−R(e1, R(e2, e3)e3)e2− R(e1, e3)(R(e2, e3)e2).
Since R(ei, ej)ek = (Aξei∧ Aξej)ek, using ( 2 . 2 ) , one can get
R(e1, e3)e1= −K13e3, R(e1, e3)e3= K13e1
R(e2, e1)e1= K12e2, R(e2, e1)e2= −K12e1 (2.7)
R(e2, e3)e2= −K23e3, R(e2, e3)e3= K23e2.
So substituting ( 2 . 7 ) and ( 2 . 2 ) into ( 2 . 5 ) and ( 2 . 6 ) , respectively we obtain ( 2 . 3 ) and
( 2 . 4 ) . Lemma 2 . 2 . Let M be a 2 - quasi - umbilical hypersurface of En+1, n ≥ 4. Then
Q(g, R)(e2, e3, e1; e1, e3) = b[c − a]e2, (2.8)
Q(g, R)(e1, e3, e2; e2, e3) = a[c − b]e1. (2.9)
Proof . Using the relation ( 1 . 7 ) we obtain
Q(g, R)(e2, e3, e1; e1, e3)
= (e1∧ e3)R(e2, e3)e1− R((e1∧ e3)e2, e3)e1 (2.10)
and
Q(g, R)(e2, e3, e2; e2, e3)
= (e2∧ e3)R(e2, e3)e2− R((e2∧ e3)e2, e3)e2 (2.11)
On 2 - Quasi - Umbilical Pseudosymmetric Hypersurfaces i − nh − te Euclide a − nSpace 4 1
So substituting ( 2 . 7 ) and ( 2 . 2 ) into ( 2 . 1 0 ) and ( 2 . 1 1 , respectively we obtain ( 2 . 8 )
and ( 2 . 9 ) .
Using Lemma 2 . 1 and Lemma 2 . 2 we have the following theorem : Theorem 2 . 2 . Let Mn be a 2 - quasi - umbilical hypersurface of
En+1, n ≥ 4. Then
Mn is proper pseudosymmetric if and only if a = b and LR= ac holds on M
n . Proof . Let Mn be a pseudosymmetric hypersurface in En+1. Then by definition one can write
(R(e1, e3) · R)(e2, e3)e1= LRQ(g, R)(e2, e3, e1; e1, e3) (2.12)
and
(R(e2, e3) · R)(e1, e3)e2= LRQ(g, R)(e2, e3, e2; e2, e3). (2.13)
Since Mn is 2 - quasi - umbilical then by Lemma 2 . 1 and Lemma 2 . 2 the equations
( 2 . 1 2 ) and ( 2 . 1 3 ) turn into respectively
b(c − a)(LR− ac) = 0 (2.14)
and
a(c − b)(LR− bc) = 0, (2.15)
respectively . Extracting the equations ( 2 . 1 4 ) and ( 2 . 1 5 ) we get
cLR(b − a) = 0. (2.16)
Since Mn is proper pseudosymmetric , it is not semisymmetric . Then the equation ( 2 . 1 6 ) gives us b = a. So the principal curvatures of Mn must be of the form (a, a, c, ..., c), which gives us
(R(e1, e3) · R)(e2, e3)e1= a2c[c − a]e2, (2.17)
Q(g, R)(e2, e3, e1; e1, e3) = a[c − a]e2, (2.18)
(R(e2, e3) · R)(e1, e3)e2= a2c[c − a]e1, (2.19)
and
Q(g, R)(e1, e3, e2; e2, e3) = a[c − a]e1. (2.20)
So from ( 2 . 1 7 ) – ( 2 . 1 8 ) and ( 2 . 1 9 ) – ( 2 . 20 ) we obtain LR = ac. This
completes the proof of the theorem . References
[ 1 ] K . Arslan , R . Deszcz and S..Yaprak , On Weyl pseudosymmetric hypersurfaces , Colloq . Math
. 72 ( 2 ) ( 1 997 ) , 353 – 36 1 . [ 2 ] B . Chen , Geometry of Submanifolds and its Applications , Sci . Univ . Tokyo , Tokyo , 1 98 1 . [ 3 ] R . Deszcz , On pseudosymmetric spaces , Bull . Soc . Math . Belg . S e´r . A 44 ( 1 ) ( 1 992 ) , 1 – 34 . [ 4 ] R . Deszcz , L . Verstraelen and S.. Yaprak
, On hypersurfaces with pseudosymmetric Weyl tensor , in Geometry and topology of submanifolds , VIII ( Brussels , 1 995 / Nordfjordeid , 1 995 ) , 1 1 1 – 1 20 , World Sci . Publ . , River Edge , NJ . [ 5 ] R . Deszcz , L . Verstraelen and S..Yaprak , On 2 - quasi - umbilical hypersurfaces in conformally
4 2 C i − hn−a ¨O
zg ¨u r
[ 6 ] R . Deszcz , L . Verstraelen and S..Yaprak , Hypersurfaces with pseudosymmetric Weyl ten - sor
in conformally flat manifolds , in Geometry and topology of submanifolds , IX ( Valenci - ennes / Lyon / Leuven , 1 997 ) , 108 – 1 1 7 , World Sci . Publ . , River Edge , NJ . [ 7 ] F . Dillen , M . Petrovic and L . Verstraelen , Einstein , conformally flat and semi - symmetric sub - manifolds satisfying Chen ’ s equality , Israel J . Math . 1 0 ( 1 997 ) , 1 63 – 1 69 . [ 8 ] C . ¨Ozgu¨r, Submanifolds satisfying
some curvature conditions imposed on the Weyl tensor , Bull . Austral . Math . Soc . 67 ( 1 ) ( 2003 ) , 95 – 10 1 .
[ 9 ] C . ¨Ozgu¨rand K . Arslan , On some class of hypersurfaces in En+1satisfying Chen ’ s equality ,
